By Leibniz' product role, we easily get that \(\psi _ja\) , \(j\in \mathbb {N}_0\) , satisfy the \(\widetilde{\mathfrak {M}}_3\) -condition uniformly in \(j\in \mathbb {N}_0\) . Consequently, Lemma REF  yields
\({\mathcal {F}}^{-1} (a\psi _j)\in {\mathcal {C}}^1(\dot{\mathbb {R}}; {\mathcal {L}}(X,Y)) \quad (j\in \mathbb {N}_0) \textrm { with }\sup _{j\in \mathbb {N}_0} \left[{\mathcal {F}}^{-1} (a\psi _j)\right]_{{\mathcal {K}}_1} <\infty .\) 
Moreover, since \(\widetilde{\mathfrak {M}}_3({\mathcal {L}}(X,Y)) \subset \widetilde{\mathfrak {M}}_2({\mathcal {L}}(X,Y))\) , by Amann's result [1]}, \({\mathcal {F}}^{-1} (a\psi _j)\in L^1({\mathcal {L}}(X,Y))\)  uniformly in \(j\) . Consequently, \((a\psi _j)(D)\) , \(j\in \mathbb {N}_0\) , are Calderón-Zygmnd operators and, by Corollary REF , the operator \(T:= a(D)_{| {\mathcal {S}}(X)}\)  satisfies the assumptions of Theorem REF \((i)\) , and, consequently, extends to an operator \({\mathcal {T}}_\Phi \)  in \({\mathcal {L}}(B^{s,q}_\Phi (X), B^{s,q}_\Phi (X))\) .
The statement on the continuity of \({\mathcal {T}}_\Phi \)  if \(q=\infty \)  follows from the fact that for the adjoint operator of \({\mathcal {T}}_{l,\Phi }\in {\mathcal {L}}(\Phi (X), \Phi (Y))\) , \({\mathcal {T}}_{l,\Phi }^* g \in \Phi ^{\prime }(X^*)\)  for all \(g\in \Phi ^{\prime }(Y^*)\) . Indeed, one can easily check that \({\mathcal {T}}_{l,\Phi }^*\)  agrees on \(\Phi ^{\prime }(Y^*)\subset [\Phi (Y)]^*\)  with the extension of the operator \((a^*\psi _l)(D)\)  to an operator in \({\mathcal {L}}(\Phi ^{\prime }(Y^*), \Phi ^{\prime }(X^*))\) , which is obtained by means of Theorem REF . Here, \(a^*(t):= a(t)^* \in {\mathcal {L}}(Y^*, X^*)\) , \(t\ne 0\) . It proves the first statement of \((i)\) .
For the second one, assume that \(a(D)\)  possesses a continuous extension on a subset \({\mathcal {G}}(X)\subset {\mathcal {S}}^{\prime }(X)\)  such that \(B^{s,q}_\Phi (X)\subset {\mathcal {G}}(X)\) . By Lemma REF  and (REF ), \(B^{s,q}_\Phi (X) \hookrightarrow {\mathcal {S}}^{\prime }(X)\)  and for every \(f\in B^{s,q}_\Phi (X)\) , \(\sum _{j=0}^N \psi _j(D) f \rightarrow f\)  as \(N\rightarrow \infty \)  both in \({\mathcal {S}}^{\prime }(X)\)  and \(B^{s,q}_\Phi (X)\)  (when \(q=\infty \)  the convergence holds in \(\sigma (B^{s,\infty }_{\Phi }(X), B^{-s,1}_{\Phi ^{\prime }}(X^*))\) -topology of \(B^{s,\infty }_{\Phi }(X)\) ), we get \(a(D)f={\mathcal {T}}_\Phi f\)  for all \(f\in B^{s,q}_\Phi (X)\) . This completes the proof of the part \((i)\) .
For the proof of \((ii)\) , first note that \(a(D)\)  is a Calderón-Zygmund operator; see Lemma REF  and considerations made in Subsection REF . Thus, the proof of this part relies on Theorem REF \((ii)\)  and uses similar arguments to those presented above.
